Since the QEC conditions in Eq. (REF ) are linear, a code that corrects perfectly for the set \(\lbrace E_{i}\rbrace \)  will also correct perfectly for any linear combination of the errors in the set. Thus, to correct for an arbitrary single-qubit error, it suffices to find codes that can correct perfectly for a unitary error basis such as the Pauli basis spanned by the single-qubit Pauli operators \(\lbrace I,X,Y,Z\rbrace \) . This in turn leads to an elegant reformulation of the theory of perfect QEC using the group theoretic algebra of Pauli operators called the stabilizer formalism [1]}, [2]}. We refer to other review articles in this special issue for further details on the stabilizer formalism and its role in quantum code construction.
